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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.dist_ref.nmp"></a><a class="link" href="nmp.html" title="Non-Member Properties">Non-Member Properties</a>
</h3></div></div></div>
<p>
        Properties that are common to all distributions are accessed via non-member
        getter functions: non-membership allows more of these functions to be added
        over time, as the need arises. Unfortunately the literature uses many different
        and confusing names to refer to a rather small number of actual concepts;
        refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">concept
        index</a> to find the property you want by the name you are most familiar
        with. Or use the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">function
        index</a> to go straight to the function you want if you already know
        its name.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h0"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.function_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">Function
        Index</a>
      </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
            Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
            Distribution Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
            complement of the probability</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
          </li>
<li class="listitem">
            entropy.
          </li>
</ul></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h1"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.concept_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">Conceptual
        Index</a>
      </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
            Distribution Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
            Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse Cumulative
            Distribution Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse Survival
            Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower Critical
            Value</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent Point Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability Mass Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">Q</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
            complement of the probability</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper Critical
            Value</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
          </li>
<li class="listitem">
            entropy
          </li>
</ul></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h2"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        The <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
        Function</a> is the probability that the variable takes a value less than
        or equal to x. It is equivalent to the integral from -infinity to x of the
        <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        For example, the following graph shows the cdf for the normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/cdf.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h3"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.ccdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
        of the Cumulative Distribution Function</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special">&lt;</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">comp</span><span class="special">);</span>
</pre>
<p>
        The complement of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a> is the probability that the variable takes a
        value greater than x. It is equivalent to the integral from x to infinity
        of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
        Function</a>, or 1 minus the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a> of x.
      </p>
<p>
        This is also known as the survival function.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        In this library, it is obtained by wrapping the arguments to the <code class="computeroutput"><span class="identifier">cdf</span></code> function in a call to <code class="computeroutput"><span class="identifier">complement</span></code>, for example:
      </p>
<pre class="programlisting"><span class="comment">// standard normal distribution object:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span> <span class="identifier">norm</span><span class="special">;</span>
<span class="comment">// print survival function for x=2.0:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">2.0</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
        For example, the following graph shows the __complement of the cdf for the
        normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/survival.png"></span>
      </p>
<p>
        See <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a> for why the complement
        is useful and when it should be used.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h4"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.hazard"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard
        Function</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">hazard</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
        of <span class="emphasis"><em>x</em></span> and distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span>

        </p></blockquote></div>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
          Some authors refer to this as the conditional failure density function
          rather than the hazard function.
        </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h5"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.chf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative
        Hazard Function</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">chf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard
        Function</a> of <span class="emphasis"><em>x</em></span> and distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span>

        </p></blockquote></div>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
          Some authors refer to this as simply the "Hazard Function".
        </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h6"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mean"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the mean of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined mean (for example the Cauchy
        distribution).
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h7"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.median"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">median</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the median of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h8"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mode"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the mode of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined mode.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h9"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
        Density Function</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">pdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        For a continuous function, the probability density function (pdf) returns
        the probability that the variate has the value x. Since for continuous distributions
        the probability at a single point is actually zero, the probability is better
        expressed as the integral of the pdf between two points: see the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a>.
      </p>
<p>
        For a discrete distribution, the pdf is the probability that the variate
        takes the value x.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        For example, for a standard normal distribution the pdf looks like this:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/pdf.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h10"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.range"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">Range</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">range</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the valid range of the random variable over distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h11"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">p</span><span class="special">);</span>
</pre>
<p>
        The quantile is best viewed as the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a>, it returns a value <span class="emphasis"><em>x</em></span> such
        that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span>
        <span class="identifier">p</span></code>.
      </p>
<p>
        This is also known as the <span class="emphasis"><em>percent point function</em></span>, or
        <span class="emphasis"><em>percentile</em></span>, or <span class="emphasis"><em>fractile</em></span>, it is
        also the same as calculating the <span class="emphasis"><em>lower critical value</em></span>
        of a distribution.
      </p>
<p>
        This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the probability lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
        if there is no finite value that has the specified probability.
      </p>
<p>
        The following graph shows the quantile function for a standard normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/quantile.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h12"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile_c"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile
        from the complement of the probability.</a>
      </h5>
<p>
        See also <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special">&lt;</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">comp</span><span class="special">);</span>
</pre>
<p>
        This is the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
        of the Cumulative Distribution Function</a>. It is calculated by wrapping
        the arguments in a call to the quantile function in a call to <span class="emphasis"><em>complement</em></span>.
        For example:
      </p>
<pre class="programlisting"><span class="comment">// define a standard normal distribution:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span> <span class="identifier">norm</span><span class="special">;</span>
<span class="comment">// print the value of x for which the complement</span>
<span class="comment">// of the probability is 0.05:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
        The function computes a value <span class="emphasis"><em>x</em></span> such that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
        <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code> where
        <span class="emphasis"><em>q</em></span> is complement of the probability.
      </p>
<p>
        <a class="link" href="../stat_tut/overview/complements.html#why_complements">Why complements?</a>
      </p>
<p>
        This function is also called the inverse survival function, and is the same
        as calculating the <span class="emphasis"><em>upper critical value</em></span> of a distribution.
      </p>
<p>
        This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the probability lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
        if there is no finite value that has the specified probability.
      </p>
<p>
        The following graph show the inverse survival function for the normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/survival_inv.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h13"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.sd"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">Standard
        Deviation</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the standard deviation of distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined standard deviation.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h14"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.support"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">support</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the supported range of random variable over the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        The distribution is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the
        smallest closed set whose complement has probability zero"</a>.
        Non-mathematicians might say it means the 'interesting' smallest range of
        random variate x that has the cdf going from zero to unity. Outside are uninteresting
        zones where the pdf is zero, and the cdf zero or unity.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h15"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.variance"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">Variance</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the variance of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined variance.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h16"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.skewness"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">Skewness</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">skewness</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the skewness of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined skewness.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h17"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">Kurtosis</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">kurtosis</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the 'proper' kurtosis (normalized fourth moment) of the distribution
        <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        kurtosis = β<sub>2</sub>= μ<sub>4</sub> / μ<sub>2</sub><sup>2</sup>
      </p>
<p>
        Where μ<sub>i</sub> is the i'th central moment of the distribution, and in particular
        μ<sub>2</sub> is the variance of the distribution.
      </p>
<p>
        The kurtosis is a measure of the "peakedness" of a distribution.
      </p>
<p>
        Note that the literature definition of kurtosis is confusing. The definition
        used here is that used by for example <a href="http://mathworld.wolfram.com/Kurtosis.html" target="_top">Wolfram
        MathWorld</a> (that includes a table of formulae for kurtosis excess
        for various distributions) but NOT the definition of <a href="http://en.wikipedia.org/wiki/Kurtosis" target="_top">kurtosis
        used by Wikipedia</a> which treats "kurtosis" and "kurtosis
        excess" as the same quantity.
      </p>
<pre class="programlisting"><span class="identifier">kurtosis_excess</span> <span class="special">=</span> <span class="char">'proper'</span> <span class="identifier">kurtosis</span> <span class="special">-</span> <span class="number">3</span>
</pre>
<p>
        This subtraction of 3 is convenient so that the <span class="emphasis"><em>kurtosis excess</em></span>
        of a normal distribution is zero.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined kurtosis.
      </p>
<p>
        'Proper' kurtosis can have a value from zero to + infinity.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h18"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis_excess"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">Kurtosis
        excess</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">kurtosis_excess</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the kurtosis excess of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        kurtosis excess = γ<sub>2</sub>= μ<sub>4</sub> / μ<sub>2</sub><sup>2</sup>- 3 = kurtosis - 3
      </p>
<p>
        Where μ<sub>i</sub> is the i'th central moment of the distribution, and in particular
        μ<sub>2</sub> is the variance of the distribution.
      </p>
<p>
        The kurtosis excess is a measure of the "peakedness" of a distribution,
        and is more widely used than the "kurtosis proper". It is defined
        so that the kurtosis excess of a normal distribution is zero.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined kurtosis excess.
      </p>
<p>
        Kurtosis excess can have a value from -2 to + infinity.
      </p>
<pre class="programlisting"><span class="identifier">kurtosis</span> <span class="special">=</span> <span class="identifier">kurtosis_excess</span> <span class="special">+</span><span class="number">3</span><span class="special">;</span>
</pre>
<p>
        The kurtosis excess of a normal distribution is zero.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h19"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdfPQ"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P
        and Q</a>
      </h5>
<p>
        The terms P and Q are sometimes used to refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a> and its <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">complement</a>
        respectively. Lowercase p and q are sometimes used to refer to the values
        returned by these functions.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h20"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.percent"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent
        Point Function or Percentile</a>
      </h5>
<p>
        The percent point function, also known as the percentile, is the same as
        the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h21"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse
        CDF Function.</a>
      </h5>
<p>
        The inverse of the cumulative distribution function, is the same as the
        <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h22"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse
        Survival Function.</a>
      </h5>
<p>
        The inverse of the survival function, is the same as computing the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the complement
        of the probability</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h23"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pmf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability
        Mass Function</a>
      </h5>
<p>
        The Probability Mass Function is the same as the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
        Density Function</a>.
      </p>
<p>
        The term Mass Function is usually applied to discrete distributions, while
        the term <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
        Function</a> applies to continuous distributions.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h24"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.lower_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower
        Critical Value.</a>
      </h5>
<p>
        The lower critical value calculates the value of the random variable given
        the area under the left tail of the distribution. It is equivalent to calculating
        the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h25"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.upper_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper
        Critical Value.</a>
      </h5>
<p>
        The upper critical value calculates the value of the random variable given
        the area under the right tail of the distribution. It is equivalent to calculating
        the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the
        complement of the probability</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h26"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival
        Function</a>
      </h5>
<p>
        Refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of
        the Cumulative Distribution Function</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h27"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.entropy"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.entropy">Entropy</a>
      </h5>
<p>
        The entropy (or differential entropy) of a continuous probability distribution
        <span class="emphasis"><em>p</em></span> is defined as
      </p>
<p>
        <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/differential_entropy.svg" width="311" height="43"></object></span>
      </p>
<p>
        Note that the "natural" properties of the differential entropy
        do not uniquely specify a log base. In the Boost library, we <span class="emphasis"><em>always</em></span>
        use the natural logarithm to compute differential entropy. This choice of
        log base for entropy is sometimes referred to as "entropy measured in
        nats". See <a href="https://doi.org/10.1109/TIT.1978.1055832" target="_top">On
        the entropy of continuous probability distributions</a> for more information.
      </p>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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